Optimal search path planning of UUV in battlefeld ambush scene

Wei Feng, Yan Ma, Heng Li, Haixiao Liu, Xiangyao Meng, Mo Zhou

Naval Research Institute, Beijing,100161, China

Keywords:Battlefield ambush Optimal search path planning UUV path Planning Probability of cooperative search

ABSTRACT Aiming at the practical application of Unmanned Underwater Vehicle(UUV) in underwater combat, this paper proposes a battlefield ambush scene with UUV considering ocean current.Firstly, by establishing these mathematical models of ocean current environment, target movement, and sonar detection, the probability calculation methods of single UUV searching target and multiple UUV cooperatively searching target are given respectively.Then, based on the Hybrid Quantum-behaved Particle Swarm Optimization (HQPSO) algorithm, the path with the highest target search probability is found.Finally,through simulation calculations,the influence of different UUV parameters and target parameters on the target search probability is analyzed, and the minimum number of UUVs that need to be deployed to complete the ambush task is demonstrated,and the optimal search path scheme is obtained.The method proposed in this paper provides a theoretical basis for the practical application of UUV in the future combat.

1.Introduction

1.1.Background

Battlefield ambush is one of the commonly used tactics in submarine combat [1].It refers to the pre-deployment of submarine in the sea area (battlefield) where the enemy target ship may appear.After target enters the sea area,the submarine will start to search for the target.Once the target is found, the torpedo is launched to destroy the target.In order to take advantage of weapons instead of forces,UUVs can be used instead of submarines to complete the above tasks.Fig.1 is a schematic diagram of the combat scene for UUV battlefield ambush.

1.2.Combat scene assumption

The battlefield ambush combat mode of UUV is mainly divided into two stages.

1) The cruiseing stage.The UUV is assigned to sail to a certain sea area hundreds of kilometers away at a constant propulsion speed (7.2-28.8 km/h) for shutdown and standby.In this process,the UUV needs to consider the influence of ocean current,obstacle collision,and navigation error to safely navigate to the designated standby position in the shortest time.

2) Approaching the enemy stage.Suppose that at a certain moment,the UUV captures the intelligence that the enemy ship target is about to cross the battlefield through the relay communication buoy.At this time, the UUV plans a path with the highest search probability through the online path planning system.Then, the UUV starts from the standby position and searches for targets along this path.Once the detection sonar of UUV finds the target ship, the UUV immediately turns into an autonomous attack state and destroys the ship at the fastest speed.In this stage, the speed of UUV in still water remains unchanged except for the final attack.

This paper only focuses on the optimal search path planning of UUV in the stage of approaching the enemy.

The following describes the UUV battlefield ambush scene in detail with reference to Fig.2.

Fig.2 is the schematic diagram of UUV in the implementation of battlefield ambush combat task.The rectangular area OABC represents the battlefield that the target ship will cross,which is actually a strait or an international channel.The point D represents the standby position of UUV,that is,the initial search position of UUV.The point E is the initial position of the target ship entering the strait.|OA| and |OC| represent the length and width of the strait respectively.Now the following assumptions are made.

Fig.1.Schematic diagram of the combat scene for UUV battlefield ambush.

Fig.2.Schematic diagram of UUV in the implementation of battlefield ambush combat task.

Hypothesis 1.The threat of obstacles in the channel is not considered, and the movement of ocean current is assumed to be steady.

Hypothesis 2.There are enough terrain matching areas distributed within the channel.For this reason, the navigation error during UUV traveling can be corrected by terrain matching assisted navigation.

Hypothesis 3.Intelligence information only provides the initial position and speed of the target ship, and the initial heading information is unknown.In addition, the heading and speed of the target ship remain unchanged during the whole process of crossing the battlefield.

Hypothesis 4.UUV can successfully destroy the target immediately after discovering the target.It is considered that the target is destroyed when it is found.Therefore, the problems need to be solved in this paper are as follows:

(1) How to obtain a path with the highest probability of

searching target for UUV when it is known that the target ship is about to enter the target sea area at a certain moment.(2) When the probability of searching target is required to reach a certain value Pf(such as 90%),how to get the optimal path search scheme,that is,how many UUVs need to be arranged in this sea area.

1.3.Brief review on path planning of UUV

The problem of underwater path planning for UUV can be divided according to the perception of environmental information.Specifically,it can be divided into offline path planning(also known as static path planning) with completely known environmental information and online path planning(also known as dynamic path planning) with unknown environmental information.

1.3.1.Research on offline path planning method

The current common offline path planning methods are mainly divided into the following four types:the first one is based on graph search algorithm,and the more typical ones are Dijkstra Algorithm[2],A*Algorithm[3],Filed D*Algorithm[4,5],etc.;the second one is based on biological intelligence algorithm, and the more typical ones are Genetic Algorithm(GA) [6], Particle Swarm Optimization Algorithm(PSO) [7], Ant Colony Algorithm(ACO) [8,9], Flower Pollination Algorithm (FPA) [10], etc.; the third one is based on sampling-based search algorithm, and the more typical ones are Probabilistic Roadmap Method (PRM) [11], Rapidly-exploring Random Tree Method (RRT) [12,13], etc.; the fourth one is other algorithms, including Artificial Potential Field Method [14,15], Visibility Graph Approach [16], Inner Normal Guided Segmentation Algorithm [17] and Homotopy Continuation Methods (HCM) [18],etc.

1.3.2.Research on online path planning method

Many offline path planning algorithms are also suitable for online path planning, such as: A* algorithm, RRT, GA, PSO, and simulated annealing approach [19], etc.In addition, there is also a class of methods that are only suitable for online path planning,such as the sliding window algorithm [20] and so on.

The current common online path re-planning methods can be divided into five types: the first one is the partially reactive path planning method,such as that in Refs.[21,22]the second one is the path planning method based on unit decomposition,such as that in Refs.[23,24];the third one is the dynamic fast search random tree method, such as that in Refs.[13,25]; the fourth one is the path planning method based on neural network learning,such as that in Refs.[26,27]; the fifth one is the path planning method based on stochastic level-set partial differential equations, such as that in Ref.[28].

In the real marine environment, there are not only static obstacles such as islands, but also many dynamic environmental factors,such as spatiotemporal current and moving obstacles[29].This requires UUV to have the ability of dynamic path planning during navigation, so it is of great significance to study the online path planning method of UUV.

1.4.Motivation and contribution

The research motivation of this paper is to efficiently obtain the optimal cooperative search path of UUV in the assumption scene of battlefield ambush combat task, and to provide a theoretical basis for the practical application of UUV in the future.Therefore, this paper establishes the probability calculation mathematical models of single UUV searching target and multiple UUV cooperatively searching target respectively,and obtains the path with the highest target search probability based on the HQPSO algorithm.Aiming at the premature convergence problem of QPSO algorithm,the HQPSO algorithm with stronger global search ability is proposed by adopting the strategies of particle average optimal position selection mutation and particle swarm individual selection mutation.The detailed algorithm design process of HQPSO can refer to our previous research results in Ref.[30].The contribution and innovation of the method proposed in this paper are as follows.

• Aiming at the battlefield ambush combat scene of UUV, the corresponding simplified mathematical model is established,and the probability calculation methods of single UUV searching target and multiple UUV cooperatively searching target are given respectively.

• This paper studies the cooperative search path planning problem of UUV battlefield ambush combat based on the HQPSO algorithm, discusses the influence of different UUV parameters and target parameters on the probability of searching target,and obtains the optimal search path scheme under different search probability requirements, which provides a theoretical basis for the reasonable deployment of UUV in the actual combat.

1.5.Organization

The rest of the paper is organized as follows.Section 2 prepares several mathematical models before planning the optimal search path; Section 3 introduces the UUV optimal search path planning method in detail; Section 4 designs several group of simulation experiments.Conclusions are made in Section 5.

2.Mathematical model preparation

To solve the problem of optimal search path planning for UUV in the battlefield ambush combat scene, the key is to design a cost function with the largest search probability as the optimization objective.Before that, several mathematical models need to be established, including the battlefield ocean current model, the target movement model and the sonar detection model of UUV.

2.1.Battlefield ocean current model

In this paper,it is assumed that the ocean current changes very slowly, that is, the speed and direction of ocean current do not change with time during the whole path search task of UUV.

In Ref.[31], a numerical equation simulation method of ocean current is proposed,which can not only simulate the ocean current movement more realistically, but also provide an ocean current map with the desired resolution according to the needs of users.Therefore,in the process of ocean current modeling,this method is used to simulate the ocean current movement.The ocean current map obtained by this method is essentially realized by superimposing multiple viscous Lamb vortices.The movement equation of a single viscous Lamb vortex can be expressed as follows[32,33]:

where u(r), v(r) and w(r) are the vortex velocity components in horizontal,longitudinal and vertical directions respectively,and κ,ξ and r0are used to describe the vortex intensity,vortex radius,and vortex center position coordinates, respectively.Fig.3 is a schematic diagram of the ocean current obtained by simulation.The grid size is 25×25 and the resolution is 1 km.It is composed of 50 Lambs superimposed,in which the coordinates of vortex center are randomly generated.κ = 54 km/h,ξ = 2 m.

Fig.3.Schematic diagram of the ocean current obtained by simulation.

2.2.Target movement model

According to the Hypothesis 3 in the combat scene, since the speed and position are known,and the course remains unchanged,it can be inferred that the area covered by ΔOEC is the area that the target ship may actually pass through when UUV knows that the target ship will start from point E in Fig.2 at a certain moment.Assuming that the course θmof the target ship obeys a uniform distribution,then the distribution range of θmis θm∊[θmin,θmax].Among them, θminand θmaxrespectively represent the minimum and maximum course angle of the target ship in the OXY Cartesian coordinate system, and take the counter clockwise as the positive.

Assuming that the speed of the target ship is Vm, and its initial position of entering the channel is (x0, y0), then the position equation of the target at any time is as follows:

2.3.Sonar detection model of UUV

Sonar is one of the commonly used equipment for searching underwater target.In the process of UUV searching for target ships,detection sonar is used to collect target information.The detection probability of sonar is a complex function affected by many factors,such as physical environment, signal power, signal-to-noise ratio,etc.The detection probability function Psof sonar is a function of detection distance rm.

There are two common detection probability models[34],one is the ideal detection model and the other is the attenuation model.Their expressions can be expressed as follows:

where,rmrepresents the distance between the UUV and the target ship, and Lsis the detection distance of sonar.

In order to simplify the problem, this paper chooses the ideal detection model to search for the target ship.

3.Optimal search path planning of UUV

In order to judge the advantages and disadvantages of different path search schemes, two methods can be used for evaluation.Method 1: Under the same constraints, compare the maximum probability of searching target; Method 2: Under the premise of successfully searching the target, compare the minimum cost of searching target (such as time expectation).The purpose of this paper is to maximize the probability of UUV searching for the target ship.Therefore, Method 1 is selected as the criterion for path evaluation.

Define the search probability Pss as the cost function for searching path.Pss is defined as follows: When 0 ≤t ≤, for any feasible search path scheme δ of UUV, the cumulative search probability at time t is Pss(δ, t), then the probability of UUV detecting the target at least once in the time period [0,t] is

It is very difficult to accurately obtain the analytical solution of Pss.Monte Carlo method can be used for approximate calculation to obtain the results.

3.1.Optimal search path planning for single UUV

3.1.1.The search probability calculation method for single UUV

For the case of single UUV searching target, the approximate calculation process of Monte Carlo method is as follows: Firstly, a large number of movement trajectories are generated randomly according to the movement model of target ship.Secondly, the behavior of UUV searching target is simulated.Finally, the search probability Pssis obtained by means of probability statistics.

Assume that the UUV starts searching targets within the time period of t ∊[0,Tmax].Since it is a continuous behavior for the UUV searching targets, this behavior is divided into Nsdiscrete events with a search interval of Δts.Then at time t,for any search path,the calculation formula of the search probability Pssis as follows:

where,t(i)represents the moment of the ith search.Nsrepresents the total number of searches.Pcs(i,j)is the search probability of the jth simulated target after the ith search.NumTrepresents the total number of target trajectories obtained by simulation.Ps(rm) is the detection probability function of sonar.rm(i,j) represents the distance between the UUV and the jth target in the ith search.

According to the ideal detection probability function of sonar,Eq.(7) can be simplified as

where,Nf(t) represents the number of targets searched at time t.

3.1.2.Steps of planning search path for single UUV

In order to solve the premature problem of QPSO algorithm in the process of convergence, mutation operation can be performed on the basis of QPSO algorithm.After the mutation operation of average optimal position and the selection mutation operation of particle swarm, Hybrid Quantum-behaved Particle Swarm Optimzation(HQPSO) is obtained.The detailed algorithm design process of HQPSO can refer to Ref.[30].

After clarifying the cost function of the search path,the HQPSO algorithm can be used to solve the optimal search path.The specific steps of the path planning method are as follows.

Step 1.Preprocess the path planning environment.According to the initial position of the target ship, NumTtarget trajectories are randomly generated.

Step 2.Read the battlefield environment information.It includes the initial position of UUV for battlefield ambushing, the sailing speed in still water and the sonar detection radius of UUV, the target ship speed, the data of ocean current field, the maximum search time, and the search time interval, etc.Among them, the maximum search time Tmaxcan be calculated according to the following equation:

Step 3.Set the algorithm parameters.It includes the number of population,the dimension of optimization variables,the maximum number of iterations and so on.

Step 4.Particle encoding and population initialization are used to obtain a series of initial search paths.In this process, as the destination of the search path planning becomes a moving target ship,the setting of control points is no longer equidistantly distributed according to the UUV starting point.Therefore, the process of particle encoding and population initialization is changed,and the new particle encoding and population initialization process is implemented according to the following method.

It can be seen from Fig.2 that assuming the initial speed of the target ship is Vm,and the maximum combined speed of the UUV is Vrmax.In the most ideal case,when the UUV and the target ship are facing each other, the shortest time Tminfor the UUV to meet the target ship is as follows:

where, Vrmaxis the maximum combined speed, Vcmaxis the maximum ocean current speed,and Vswis the UUV sailing speed in still water.

Therefore,the longest distance Lmaxof UUV sailing along the Xaxis is

Due to the backtracking situation of UUV in the process of searching for the target,five control points are set in all the search paths planned in this article.Among them, the first four control points are equidistantly distributed, and the abscissa of the last control point is not fixed, but is regarded as a variable to be optimized.According to the principle of the HQPSO algorithm, the dimension of the particle is Dd=6 at this time.Assuming that there are N particles, the coding of the ith particle xiis as follows:

Therefore,the initialization of the individual particle swarm can be realized by a random number generator according to Eq.(15).

Step 5.The HQPSO algorithm is used to update the population.

Step 6.Calculate the cost of each search path according to the cost function defined by Eq.(10), and calculate the individual optimal extremum and the global optimal extremum of each particle.

Step 7.Recalculate the individual optimal extremum and global optimal extremum of each particle for the updated population.

Step 8.Repeat Step 5 to Step 7 until the maximum number of iterations is reached.

Step 9.End the iteration.Output the coordinates of optimal control point and generate the optimal search path.The algorithm flow chart of planning search path for single UUV is shown in Fig.4.

3.2.Optimal search path planning for multiple UUV

Because a single UUV often cannot ensure that the target is completely searched, it is necessary to study the path planning problem of multiple UUVs for collaborative search.

3.2.1.The search probability calculation method for multiple UUV

For the situation where multiple UUVs are searching for targets at the same time, due to the problem of repeated searching for targets in the collaborative search of multiple UUVs, the collaborative search probability Pcomcannot be directly obtained by simply accumulating the search probability of each single UUV.

Assuming that there are n UUVs setting out to search for the target ship at the same time.NumTtarget trajectories are randomly generated.At the moment of t, if the ith UUV searches for the jth target, then record Ct(i, j) = 1, otherwise record Ct(i, j) = 0.The definition of the marking function H(t, j) is as follows:

Fig.4.The algorithm flow chart of planning search path for single UUV.

The above equation indicates that when all UUVs do not search for the jth target,the value of H(t,j)is 1;and when at least one UUV searchs for the jth target,the value of H(t,j)is 0.If Nr(t)is the total number of unsearched targets at the moment of t,then Nr(t)can be calculated by the following equation:

3.2.2.Steps of planning search path for multiple UUV

The steps of planning search path for multiple UUV are similar to that of single UUV, but the difference lies in the process of the coding and initialization of the particle swarm.In the search path planning of multiple UUVs,the variable range of the control points should be determined according to the number of UUVs.The general idea is as follows: Firstly, the search area is divided into equal areas according to the number of UUVs, then the change range of the path control point position is determined according to the divided area corresponding to each UUV, and finally the search range of each dimension variable of the particle in the HQPSO algorithm is obtained.The particle initialization process in the case of multiple UUVs will be described in detail with reference to Fig.5.

Fig.5.Area division diagram when three UUVs search for targets.

Fig.5 is the area division diagram when three UUVs search for targets.It can be seen that in the case of searching target by three UUVs, the battlefield area is divided into three triangles of equal area, where D and F are the three division-points on the line segment OC respectively.Connect CE,DE,FE,and OE respectively to get the straight lines l1-l4.The equation of the ith line liis as follows:

According to the setting rules of the control points in a single UUV, the abscissa of the first four control points are equidistantly distributed.Then their corresponding linear equations is: x = xi,i=1,2,3,4.Therefore,the coordinates of any intersection point can be obtained according to the two linear equations.In Fig.5,y1,min(1)and y1,max(1) are respectively the intersection coordinates of line x=x1and lines l1∶y=k1x+b1,l2∶y=k2x+b2.They respectively correspond to the upper and lower limits of the ordinate for the first control point that determines the searched path of UUV1.According to the same method,the variation range of the ordinate for each control point can be solved in turn.In this way, the initialization of the first four dimensional variables in the particle can be realized.According to the specific combat scene in this paper, the value range of the fifth-dimensional variable of all particles is:0≤xi,5≤35.6.As for the ordinate variable of the fifth control point corresponding to the ith UUV search path (the sixth-dimensional variable in the particle), the variation range can be obtained according to the following equation:

Through the above processing, the initialization process of the HQPSO algorithm in the case of multiple UUVs can be realized.Compared with the single UUV, the following algorithm steps are the same except that the calculation formula of cost function needs to be changed to Eq.(18).For this reason,no further explanation is given here.

4.Simulation and result analysis

4.1.Results and analysis of search path planning for single UUV

Initial simulation conditions: The initial position of the UUV battlefield ambush is(0,50).The initial position of the target ship is(100, 50).The sailing speed of UUV in still water Vsw= 10.8 km/h.The ship speed Vm=33.33 km/h.The number of target movement trajectories NumT= 1000.The time interval of sonar detection Δts=50 s.The maximum search time Tmax=12,100 s.The detection radius of sonar Ls= 5 km.The number of particle population N=100.The particle dimension Dd=6.The maximum number of iterations MaxDT=100,Pa=Pb=0.5.The simulation calculation is run independently for 50 times,and the average search probability,the maximum search probability and the standard deviation of the search probability are recorded respectively.

The path planning results are shown in Figs.6 and 7.Fig.6 shows the optimal search path of UUV.Comparing to Fig.3,it can be found that when sailing along this path,the UUV happens to be advancing in the direction of the maximum ocean current speed.This shows that UUV chooses to approach the target as fast as possible when searching for the target.The purpose of this navigation is to reduce the spread range of the target and increase the density of the target in per unit area, so that the UUV has a greater probability of searching the target.In addition,the“*”in the figure indicates the location of the target when it was discovered, and it can be seen that the area where the target was searched is very concentrated.This shows that due to the low navigation speed of the UUV, the time for the target to cross the detection range is very short,which makes it difficult for the UUV to find the target again in the remaining search time range.Fig.7 is the convergence graph of the average search probability with the number of iterations obtained after 50 independent runs.It can be seen that the search probability increases with the number of iterations and finally converges to a stable solution.The above results show that the HQPSO algorithm always takes the maximum target search probability as the optimization objective in the optimization process,which can find the path with the largest search probability for UUV.This also verifies the effectiveness of the path planning method based on the HQPSO algorithm.

In order to further clarify the influence of different parameter conditions on the search probability,the following simulations are carried out on conditions of different navigation speeds for UUV in still water, different detection radius of UUV sonar, different navigation speeds of target ship, and different initial position of target ship.

4.1.1.Influence of different UUV navigation speed on search probability

Suppose that the navigation speeds Vmfor UUV in still water are respectively 7.2 km/h,10.8 km/h,14.4 km/h,18.0 km/h, 21.6 km/h,25.2 km/h,28.8 km/h,and other simulation conditions are the same as those in subsection 4.1.The simulation calculation under each condition is run independently for 50 times,and the average search probability, the maximum search probability and the standard deviation of the search probability under each simulation condition are recorded respectively.The statistical results are shown in Table 1.

Fig.6.Optimal search path of UUV when Vsw = 10.8 km/h, Vm = 33.33 km/h,Ls = 5 km.

Fig.7.Convergence diagram of average search probability when Vsw = 10.8 km/h,Vm = 33.33 km/h, Ls = 5 km.

Table 1 shows the corresponding search probabilities under different navigation speeds for UUV in still water.It can be seen from the table that as the navigation speed for UUV in still water increases successively, the search probability increases significantly.When the navigation speed in still water changes from 7.2 to 28.8 km/h,the average search probability increases by 18.5%.Fig.8 is the diagram of optimal search path when Vm= 28.8 km/h, and Fig.9 is the convergence diagram of average search probability when Vm= 28.8 km/h.By comparing Fig.6 with Fig.8, it can be seen that when the UUV sailing speed in still water is Vm=28.8 km/h, the UUV also approaches the target at the fastest speed first.

However, in Fig.8, a roundabout search section appears in the path of UUV, and the area where the target is searched is also concentrated and scattered on this section in path.This shows that when the speed of UUV increases, the time range for the UUV to detect the target becomes longer,which improves the probability of the target being searched.Fig.9 corresponds to the convergence of the average search probability at this time.It shows that when Vm=28.8 km/h the average search probability of the target finally converges to 33.4%.

Fig.9.Convergence diagram of average search probability when Vm = 28.8 km/h.

4.1.2.Influence of different UUV sonar detection radius on search probability

Suppose that the detection radius Lsof UUV are respectively 2 km, 3 km, 4 km, 5 km, 6 km, 7 km, 8 km, and other simulation conditions are the same as those in subsection 4.1.The simulation calculation under each condition is run independently for 50 times,and the average search probability, the maximum search probability and the standard deviation of the search probability under each simulation condition are recorded respectively.The statistical results are shown in Table 2.

Table 2 shows the corresponding search probability under different detection radius of UUV sonar.It can be seen from the table that as the detection radius of UUV sonar increases successively, the search probability increases.Fig.10 shows the optimal search path of UUV when the sonar detection distance is 8 km.Comparing with Fig.6,it can be seen that the two search paths are basically the same, which illustrates the stability of the HQPSO algorithm for searching the optimal solution.However,in Fig.10,due to the increase of sonar detection distance, the coverage area of each detection becomes larger,so the probability of the target being searched is also greater.Fig.11 corresponds to the convergence of the average search probability at this time.When Ls= 8 km, the average search probability of the target finally converges to 25.9%stably.

4.1.3.Influence of different target navigation speed on search probability

Suppose that the navigation speeds Vmof target are respectively 33.33 km/h, 37.04 km/h, 40.74 km/h, 44.44 km/h, 48.15 km/h,51.85 km/h, 55.56 km/h, and other simulation calculation under each condition is run independently for 50 times, and the average search probability, the maximum search probability and the standard deviation of the search probability under each simulation condition are recorded respectively.The statistical results are shown in Table 3.

Table 3 shows the corresponding search probability under different sailing speeds of target.It can be seen from the table that as the navigation speed of target increases, the target search probability of UUV decreases.Fig.12 shows the optimal search path whenthe navigation speed of target is 55.56 km/h.Comparing with Fig.6,it can be seen that the search path of the UUV becomes shorter at this time, and the area where target is searched appears near the starting point of the UUV.The main reason for this is that when the navigation speed of the target increases, on the one hand, it will make the navigation distance longer at the same time and the dispersion range of the target larger,which will lead to the decrease of the target density in per unit area;on the other hand,it will make the time of the target crossing the sonar detection range shorter,so the probability of the target being searched will be lower.Fig.13 corresponds to the convergence of the average search probability at this time.When Vm=55.56 km/h,the average search probability of the target is 13.5%.

Table 1 The relationship between navigation speed and search probability of UUV in still water.

Table 2 The relationship between search radius and search probability of UUV.

Fig.10.Optimal search path of UUV when Ls = 8 km.

Fig.11.Convergence diagram of average search probability when Ls = 8 km.

Fig.12.Optimal search path of UUV when Vm = 55.56 km/h.

Fig.13.Convergence diagram of average search probability when Vm = 55.56 km/h.

Table 3 The relationship between navigation speed of target and search probability.

Table 4 The relationship between initial positions of target and search probability.

Fig.15.Optimal search path of UUV when the initial position of the target is(100,100).

Fig.17.Convergence diagram of search probabilities in the case of two UUVs.

Fig.18.Optimal search paths in the case of three UUVs.

Fig.16.Optimal search paths in the case of two UUVs.

4.1.4.Influence of different target initial positions on search probability

Suppose that the initial positions of target are respectively(100,0), (100,25), (100,50), (100,75), (100,100), and other simulation conditions are the same as those in subsection 4.1.The simulation calculation under each condition is run independently for 50 times, and the average search probability, the maximum search probability and the standard deviation of the search probability under each simulation condition are recorded respectively.The statistical results are shown in Table 4.

Fig.19.Convergence diagram of search probabilities in the case of three UUVs.

Table 4 shows the corresponding search probability under different initial positions of target.It can be seen from the table that the different initial positions of target have little influence on the search probability, and the maximum probability difference between them is only 1.1%.Figs.14 and 15 are the optimal search paths of UUV when the target initial position coordinates are(100,75) and (100,100) respectively.It can be found that the two paths are basically similar, and the average search probability difference is only 0.7%.

Fig.20.Optimal search paths in the case of six UUVs.

Fig.21.Convergence diagram of search probabilities in the case of six UUVs.

Through the analysis of the influence of the above different conditions on the UUV target search probability, the following conclusions can be obtained.

(1) The higher the navigation speed of UUV in still water, the higher the probability of target being searched.

(2) The larger the detection radius of UUV sonar,the higher the probability of target being searched.

(3) The higher the navigation speed of target, the lower the probability of target being searched by UUV.

(4) When the initial position of the target is different, the probability of the target being searched remains basically unchanged.

4.2.Results and analysis of search path planning for multiple UUV

In order to solve the second problem raised in the combat scene assumption: When the probability of the target being searched reaches a certain value(for example,90%),how many UUVs need to be deployed in the sea area?In this paper,the search path planning problem in the case of multiple UUVs is simulated and analyzed.

Fig.22.Optimal search paths in the case of seven UUVs.

Fig.23.Convergence diagram of search probabilities in the case of seven UUVs.

Initial simulation conditions:The initial position of target ship is(100,50).The sailing speed of UUV in still water Vsw=10.8 km/h.The ship speed Vm=33.33 km/h.The number of target movement trajectories NumT= 1000.The time interval of sonar detection ΔTs= 50 s.The maximum search time Tmax= 12,100 s.The detection radius of sonar Ls= 5 km.The number of particle population N=100.The particle dimension Dd=6.The maximum number of iterations MaxDT= 100, Pa= Pb= 0.5.The simulation calculation is run independently for 50 times.The initial position of UUV is set according to the number of UUVs,and the search area is divided according to the principle of equal area.For example, the initial positions when two UUVs are arranged are(0,75),(0,25),and the initial positions when three UUVs are arranged are (0,16.6),(0,50), (0,83.3).Other situations can be deduced by analogy.

Figs.16-23 respectively correspond to the simulation results when 2 UUVs, 3 UUVs, 6 UUVs and 7 UUVs are arranged.Table 5 records the average collaborative search probability, the maximum collaborative search probability, and the standard deviation of collaborative search probability that calculated under different UUV numbers.

It can be seen from Table 5 that with the number of UUVs increasing, the probability of collaborative search increases by more than 10%.When six UUVs are arranged,the UUVs can search for the target ship with a probability close to 90%.When seven UUVs are arranged, it is basically ensured that the UUVs can fully search for the target ship.As can be seen from Figs.16,18,20 and 22,the total range covered by the search path becomes larger with the number of UUVs increasing.Each search path does not interfere with each other.This maximizes the coverage of the target distribution space and ensures that the probability of the target being detected increases with the number of UUVs.From Figs.17,19, 21 and 23, it can be seen that the average cooperative search probability increases with the number of iterations,and it can eventuallyconverges to a stable solution.This shows that the HQPSO algorithm is also effective and feasible in the case of multiple UUVs in the optimal path planning problem with the largest collaborative search probability as the goal.According to the simulation results,if the probability of the target being searched reaches Pf= 90%, the optimal search scheme is to arrange seven UUVs.In this case, as long as the target ship enters the battlefield,it must be searched by the UUV.

Table 5 The relationship between UUV quantities and search probability.

5.Conclusions

Based on the combat scene assumption of battlefield ambush for UUV, this paper establishes the probability calculation mathematical models of single UUV searching target and multiple UUV cooperatively searching target respectively, and obtains the path with the highest target search probability based on the HQPSO algorithm.The method in this paper can provide an optimal search path scheme that meets the probability requirements of the target being searched, which will provide a theoretical basis for the practical application of UUV in the future warfare.

The conclusions obtained through the simulation and quantitative analysis in this paper are highly consistent with the practical application experience, which indirectly proves the effectiveness and applicability of the path planning method proposed in this paper from the perspective of application.

Although the method proposed in this article has a certain guiding effect on the actual warfare,there are still some researches that need to be improved and perfected in the future.

(1) Path planning in three-dimensional space UUV should also have vertical maneuverability and detection capabilities during navigation.It can realize the battlefield surveys in three-dimensional space through vertical movement.For this reason,the path planning problem in three-dimensional space should be further considered in the follow-up research.

(2) Collaborative path planning for multiple UUV under the condition of space-time synchronization.The collaborative path planning models of single UUV and multiple UUVs established in this paper are limited to the optimal search path problem in a given combat scene.

In the future warfare of Underwater Unmanned system, more complex optimization allocation of time and space resources will be involved, which requires multiple UUVs to be able to perform path planning in complex combat scene.Therefore,in the follow-up research, we should further study the collaborative path planning for multiple UUV under the condition of time and space synchronization in complex scene.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.